Cycle Partitions in Dense Regular Digraphs and Oriented Graphs

A conjecture of Jackson from 1981 states that every \(d\)-regular oriented graph on \(n\) vertices with \(n\leq 4d+1\) is Hamiltonian. We prove this conjecture for sufficiently large \(n\). In fact we prove a more general result that for all \(\alpha>0\), there exists \(n_0=n_0(\alpha)\) such that every \(d\)-regular digraph on \(n\geq n_0\) vertices with \(d \geq \alpha n \) can be covered by at most \(n/(d+1)\) vertex-disjoint cycles, and moreover that if \(G\) is an oriented graph, then at most \(n/(2d+1)\) cycles suffice.